Suppose $X$ is a metric space that is covered by a countable collection of totally bounded subsets of X. Show that X is separable.
Let's say $C_n$ is the collection. If $X$ is covered by countable collection of totally bounded subsets of $X$ then the union $\bigcup C_n$ is countable and totally bounded. The union covers the set, $X \subseteq \bigcup C_n$ and so $X$ is totally bounded and thus separable.
Any help would be appreciated.